Numerical experiments on EC’ model

We have shown in Section 3.2.4 that DGImpl, compared to the pdepe, is the best numerical solver of the ETO model. Since the EC’ model can be reduced to the ETO model, for K_r = 0, then we realise here some numerical experiments to compare the performance of DG_Impl and pdepe on the EC’ model. By default, we set the parameters K₀, D⁺, D⁻, D, K_A, K_r and ˜C_A⁰total as follow

K0 = 20, D⁺= 1, Kr = 10000, D⁻= KA= D = 5 and ˜C_A⁰total = 1. (3.95)

Numerical simulation of the concentration profile of species

In order to validate our space discretization, we simulate the concentration profile of A, B, Q, Q⁺ and Q + Q⁺, using the DG method combined with the Implicit Euler method. If the species Q and Q⁺ have the same diffusion coefficient, then according to (3.79), (3.80) and the boundary conditions from (3.19) to (3.21), the sum ˜S of the dimensionless concentration of the species Q and Q⁺ is governed by (3.58), as for ETO model. We use DG_Impl to solve (3.85) (obtained with the DG discretization) subject to (3.87) for

K₀ = 20, D⁺ = 1, K_r = 10000, D⁻ = K_A = D = 5 and ˜C_A⁰total = 1.

For all z and ˜t, we respectively plot the dimensionless concentrations ˜C_Q, ˜C_Q⁺, C˜_B, ˜C_A and ˜S in Fig 3.19(a), Fig 3.19(b), Fig 3.19(c), Fig 3.19(d) and Fig 3.19(e).

Note that in Fig 3.19(e), we have ˜S = 1, ∀x, ˜t as expected. Fig 3.5(a), Fig 3.5(b), Fig 3.19(a) and Fig 3.19(b) show that the diffusion layer of the species Q and Q⁺ is larger in ETO mechanism compared to EC’ mechanism. This is expected since the specie Q is quickly refurbished by the catalytic reaction in the EC’ mechanism.

(a)

(b)

(c)

(d)

(e) Dimensionless concentration profile ˜CQ+ ˜C_Q+

Figure 3.19: Dimensionless concentration profile of the species A, B, Q and Q⁺ for K₀ = 20, D⁺ = 1, K_r = 10000, D⁻ = K_A = D = 5 and ˜C_A⁰total = 1. For all z and ˜t, we respectively plot the dimensionless concentrations ˜CQ, ˜C_Q⁺, ˜CB, C˜_A and ˜S in (a), (b), (c), (d). We also plot in (e) the dimensionless concentration S = ˜˜ CQ+ ˜C_Q⁺. Note that in (e), we have ˜S = 1, ∀x, ˜t as expected.

We compare the convergence and the efficiency of the solvers pdepe and DGImpl

with respect to the time and space discretization, while computing the concentration profile of the species A, B, Q and Q⁺ at the final time ˜t₂.

Firstly, we examine the convergence and the efficiency with respect to the time discretization. To do so, we compute the dimensionless concentration profile of the species A, B, Q and Q⁺ at the final time, using pedpe and DG_Impl, for the each time step ∆t_i = 2ⁱ∆t₀, for i = 0, · · · , 4. By considering the concentration profile associated to the finest time step ∆t0as exact concentration, we compute the relative error associated to the time step ∆t_i, i ∈ {1, · · · , 4} by the relative error E_i^1,r decrease with the time step and the curves associated to the both solvers are parallel. We can conclude that the solvers pedpe, DG_Impl converge in time and have the same order of convergence with respect to the time step. Note that in Fig 3.8(b), for a given relative E⁰, we have

T_pdepe⁰ > T_DG⁰

Impl, (3.97)

where T_r⁰ is the CPU time spends by the solver r = pdepe, DG_Implfor the computa-tion of ˜C_A, ˜C_B, ˜C_Q and ˜C_Q⁺ at the time ˜t = ˜t₂ such that E_i^1,r

0 = E⁰ . Thus DG_Impl is the most efficient solver with respect to the time discretization.

(a) (b)

Figure 3.20: Convergence and the efficiency with respect to the time dis-cretization while computing ˜C_A, ˜C_B, ˜C_Q and ˜C_Q⁺ with the solvers pdepe and DG_Impl. The convergence and the efficienciness in this case are respectively il-lustrated by plotting E_i^1,r vs ∆ti in (a) and E_i^1,r vs CPU time in (b) for i = 1, · · · , 4.

(a) shows that the solvers pedpe and DG_Impl have the same order of convergence with respect to the time step, but DG_Impl is more accurate. (b) shows that DG_Impl is more efficient, with respect to the time step, to compute the concentration of the species Q and Q⁺ at the final time ˜t₂.

Finally, we examine the convergence and efficiency with respect to the mean of the space step by computing the concentration profile of the species A, B, Q and Q⁺ at the final time, using pedpe and DG_Impl, for the each partition Ω_i, for i = 1, · · · , 5. The partitions Ω_i , for i = 2, · · · , 5 are obtained by splitting each element of Ω₁ into i equidistant elements, for a given partition Ω₁. By considering the concentration profile associated to the finest space step H₅ of the partition Ω₅ as exact concentration, we compute the relative error associated to the mean of the space step, H_i of the partition Ω_i, i ∈ {1, · · · , 4}, as follow the relative error E_i^2,r decrease with the mean of the space step H_i for all r = pdepe, DGImpl. It also shows that E_i^2,pdepe > E_i^2,DGImpl for a given time step ∆t.

Therefore the solver DG_Impl is more accurate than pdepe with respect to the space discretization. Fig 3.21(b) shows that for a given relative E⁰, we have T_pdepe⁰ >

T_DG⁰ _Impl, where T_r⁰ is the CPU time spends by the both solvers for the computation of the dimensionless concentration ˜C_A, ˜C_B, ˜C_Q and ˜C_Q⁺ such that E_i^8,r

0 = E⁰ at the time ˜t = ˜t₂. Thus DG_Impl is the most efficient solver with respect to the space discretization.

(a) (b)

Figure 3.21: Convergence and the efficiency with respect to the space dis-cretization while computing ˜C_A, ˜C_B, ˜C_Q and ˜C_Q⁺ with the solvers pdepe and DG_Impl. The convergence and the efficiency are respectively illustrated by plotting E_i^1,r vs H_i in (a) and E_i^1,r vs CPU time (b). (a) shows that the solvers pedpe and DG_Impl have the same order of convergence with respect to the space step. (b) shows that DG_Impl is more efficient, with respect to the space step, to compute the concentration of the species A, B, Q and Q⁺ at the final time ˜t₂.

Numerical simulation of the dimensionless voltammogram

We compare the voltammograms obtained using DG_Impland Matlab’s solver pdepe.

This comparison is illustrated in Fig 3.22. We plot in Fig 3.22(a) both simulated voltammograms and the zoom of a portion of their curve. Note that in Fig 3.22(a), the voltammogram obtained with pdepe present some oscillations. We plot in Fig 3.22(b), the absolute value of the difference between both voltammograms. It shows that despite the oscillation of the voltammogram obtained with pdepe, both voltammograms are almost the same since we have

G_pdepe(t_n) − G_DGImpl(t_n)  

< 0.12,

where Gpdepe and GDG_Impl respectively represent the current obtained with pdepe and DG_Impl.

(a) (b)

Figure 3.22: Comparison of the dimensionless voltammograms obtained by DG_Impl and Matlab’s solver pdepe for the EC’ model with K₀ = 20, D⁺ = 1, D⁻ = D = K_A = 5, K_r = 10000 and ˜C_A⁰total = 1. We plot in (a) both voltam-mograms and a the highlight of their portion. See in (a) that the voltammogram obtained with pdepe present some oscillations but the one obtained with DG_Impl doesn’t. We plot in (b) the absolute value of the difference between both voltam-mograms. it shows that despite the oscillation of the voltammogram obtained with pdepe, both voltammograms are almost everywhere the same.

We now examine the convergence and the efficiency of the solvers pedpe and DG_Impl with respect to the time and space discretization, while computing the dimensionless current. To do so, we firstly simulate the dimensionless voltammogram using the solvers pedpe and DG_Implfor all time step ∆t_i = 2ⁱ∆t₀, for i ∈ {0, · · · , 4}.

By considering the dimensionless current associated to the finest time step ∆t0 as the exact dimensionless current, we compute the relative error associated to the time step ∆t_i = 2ⁱ∆t₀, for i ∈ {1, · · · , 4} using (3.69). For ∆t₀ = 0.0269, we plot E_i^4,r vs

∆t_i in Fig 3.23(a) and E_i^4,r vs CPU time in Fig 3.23(b) for i ∈ {1, · · · , 4}. Note that in Fig 3.23(a), the relative error E_i^4,r tend to zeros with the time step for all r = pdepe, DG_Impl. therefore both solvers converge with respect to the time step while simulating the voltammogram. Fig 3.23(b) shows that for a given value E₀, DG_Impl compared to pdepe will spend less time to simulate the dimensionless voltammogram with a relative error equal E₀. Thus DG_Impl is more efficient, with respect to the time discretization, than pdepe to simulate the dimensionless voltammogram.

(a) (b)

Figure 3.23: Convergence and the efficiency of pdepe and DG_Impl with respect to the time discretization while simulating the dimensionless voltammogram. We plot E_i^4,r vs ∆t_i in (a) and E_i^4,r vs CPU time in (b) for i ∈ {1, · · · , 4}. Note that in (a), the relative error E_i^4,r tend to zeros with the time step for both solvers, meaning that the solvers converge with respect to the time step while simulating the voltammogram. (b) shows that for a given value E₀, DG_Impl compared to pdepe will spend less time to simulate the dimensionless voltammo-gram with a relative error equal E0. Thus DGImpl is more efficient with respect to the time discretization while simulating the dimensionless voltammogram.

Finally, we examine the convergence and the efficiency of pdepe and DGImpl with respect to the mean of the space steps while simulating the dimensionless voltammogram We simulate the dimensionless voltammogram using the solvers pedpe and DG_Impl for all partition Ω_i, i = 1, · · · , 5. The partitions Ω_i , for i = 2, · · · , 5 are obtained by splitting each element of Ω₁ into i equidistant ele-ments, for a given partition Ω₁. By considering the dimensionless voltammogram associated to the finest mean space step H₅ of the partition Ω₅ as exact dimen-sionless voltammogram, we compute the relative error E_i^5,r associated to the mean space step H_i of the partition Ω_i, i ∈ {1, · · · , 4} using (3.70). For H₅ = 0.0582, we plot E_i^5,r vs H_i in Fig 3.24(a) and E_i^5,r vs CPU time in Fig 3.24(b) for i = 1, · · · , 4.

Note that in Fig 3.24(a), the relative error E_i^5,r tend to zeros with the mean space step for the solvers pdepe and DG_Impl. Then the both solvers converge with re-spect to the mean space step while simulating the dimensionless voltammogram.

Fig 3.14(b) shows that for a given value E₀, DG_Impl compared to pdepe will spend less time to simulate the dimensionless voltammogram with a relative error equal E₀. Thus DG_Impl is more efficient than pdepe while simulating the dimensionless

voltammogram.

(a) (b)

Figure 3.24: Convergence and the efficiency of pdepe and DG_Impl with respect to the mean of the space steps while simulating the dimensionless voltammogram. We plot E_i^5,r vs H_i in (a) and E_i^5,r vs CPU time in (b) for i = 1, · · · , 4. Note that in (a), the relative error E_i^5,r tend to zeros with the mean space step for pdepe and DG_Impl. Then both solvers converge with respect to the mean space step while simulating the dimensionless voltammogram. (b) shows that for a given value E₀, DG_Impl compared to pdepe will spend less time to simulate the dimensionless voltammogram with a relative error equal E₀. Thus DG_Impl is more efficient with respect to the space discretization while simulating the dimensionless voltammogram.

3.4 Summary

In this chapter, we have developed step by step a novel numerical method, based on the DG space discretization, to solve the governing equations of the cyclic voltam-metry models and simulate its signal response. The finite DG space is constructed with the orthonormal shifted Legendre polynomials, which reduced the mass matrix to the identity matrix.

To that end, we initially investigate in Section 3.2, the DG method for the ETO models, since it has an analytical results from Aoki et al.[9], that can be used to validate the numerical solution. We introduced the dimensionless parameters to transform the linear governing equations into the dimensionless one, which is a time dependent PDEs. Using the DG space discretization, we transformed the dimensionless governing equations into an ODEs. The obtained ODEs is then solved with implicit Euler or Exponential time differencing method. We performed various

numerical experiments in Section 3.2.4, to validate and compare our solver with Matlab’s solver pdepe. Contrary to expectations, while combine with the DG spatial discretization, the standard implicit time integrator out performs the methods such as the ETD method and adaptive time stepping method ode15s. We also see that the DG method performs better than the standard FE method.

We finally investigate in Section 3.3, the DG method for the EC’ models, which is described by a non linear equations. The same process as for ETO model is used to solve the dimensionless equations of the EC’ model. Once again, various numer-ical experiments performed in Section 3.3.4, have shown that our novel numernumer-ical method, constructed by the combination of DG space discretization and Euler time discretization, is much more efficient to investigate EC’ model.

Now that an efficient solver is proposed to simulate the signal response of the cyclic voltammetry for some given parameters, we will derive in the next chapter the adjoint equation of the inverse model. The adjoint equation will allow us the compute efficiently the gradient necessary to invert the forward model, while using the gradient descent algorithm.

One dimension Inversion of Cyclic Voltammetry models

Contents

4.1 Introduction to the inverse problem . . . 90 4.2 Adjoint method for ODE-constrained optimization . . . 92 4.3 Numerical inversion of the ETO model . . . 96 4.4 Numerical inversion of the EC’ model . . . 109 4.5 Summary . . . 122 Central to this chapter is the construction of the numerical process to find all the parameters of the cyclic voltammetry models ETO and EC’ (investigated in the previous chapter) from the results of the measurement of the current. This approach seeks the quantitative agreement of the modelling of the cyclic voltammetry with results of the experimental measurement of the current. To achieve this goal, the following steps are taken: we give in Section 4.1 an overview of inverse theory and specifically define the inversion problem of cyclic voltammetry models. Then we show how the adjoint method is used to solve our inversion problem. Finally we combine the DG method, Implicit Euler method and the adjoint method to respectively invert ETO and EC’ model in Section 4.3 and Section 4.4. The results of these numerical inversions are then compared to the results obtained using the MATLAB code pdepe with the finite difference method (FD) used to compute the

gradient.

The novel contribution here is the inversion of the cyclic voltammetry models with the combination of the DG discretization, Impl time integrator and the adjoint method. The performance of our numerical inversion method is tested against a commercial MATLAB code for the inversion of synthetic data.

4.1 Introduction to the inverse problem

Generally, the purpose of collecting data is to gain meaningful information about a physical system or phenomenon of interest. However, in many situations the quan-tities that we wish to determine, which we call model parameters, are different from the quantities we are able to measure, which we call data. If the data depends, in some way, on the model parameters, then the data at least contains some informa-tion about the model parameters.

The forward problem is defined as mapping of model parameters in a functional space P, typically a Banach or Hilbert space, to the space of data G, typically another Banach or Hilbert space. It can be mathematically described as follow

G = F(p), for p ∈ P, G ∈ G, (4.1)

where F is a known function, p represent the model parameters and G is the mea-sured data [144, 21].

Starting with knowledge of the measured data G in G, the problem of trying to reconstruct the model parameters p in P is called an inverse problem such that (4.1) holds or an approximatively holds due to the error in the measurement. However, the resolution of an inverse problem is capable of doing more than estimating model parameters. It can also be used to bound the range of acceptance of model pa-rameters, estimate the uncertainties in the model papa-rameters, to do the sensitivity analysis of the data or to find what kind of data are best suited to determine the set of model parameters, see for example [210, 119, 7, 222, 209] for more details. If the inverse problem does not have a unique solution p in P, it is called an ill-posed

inverse problem [162].

Inverse problems are frequently used in a large variety of problems in sciences:

curve fitting, Acoustic Tomography, X-ray Imaging, factor analysis [165], Gravita-tional waves [155], satellite navigation [112, 121], earthquake response signals [183].